Note on Tautological Classes of Moduli of K3 Surfaces

Abstract

In this note we prove some cycle class relations on moduli of K3 surfaces. This note deals with a few properties of tautological classes on moduli spaces of K3 surfaces. Let M2d denote a moduli stack of K3 surfaces over an algebraically closed field with a polarization of degree 2d prime to the characteristic of the field. The Chern classes of the relative cotangent bundle 1 /M of the universal K3 surface X2d = X define classes t1 and t2 in the Chow groups CH i(X2d) of the universal K3 surface over M2d. The class t1 is the pull back from M2d of the first Chern class v = c1(V ) of the Hodge line bundle V = �∗( 2 /M ). We use Grothendieck-Riemann-Roch to determine the push forwards of the powers of t2. These are powers of v. We then prove that v 18 = 0 in the Chow group with rational coefficients ofM2d We show that this implies that a complete subvariety of M2d has dimension at most 17 and that this bound is sharp. These results are in line with those for moduli of abelian varieties. There the top Chern classg of the Hodge bundle vanishes in the Chow group with rational coefficients. The idea is that if the boundary of the Baily-Borel compactification has co-dimension r then some tautological class of co-dimension r vanishes. Our result means that v 18 is a torsion class. It would be very interesting to determine the order of this class as well as explicit representations of this class as a cycle on the boundary, cf., (2, 3). 2. The Moduli Space M2d Let k be an algebraically closed field. We consider the moduli space M2d of polarized K3 surfaces over k with a primitive polarization of degree 2d. This is a 19-dimensional algebraic space. Over the complex numbers we can describe it as an orbifold quotient 2d\2d, where 2d is a bounded symmetric domain and 2d an arithmetic subgroup of SO(3,19) obtained as follows. Consider the lattice U 3 ⊕E 2 8 , where U denotes a hyperbolic plane and E8 the usual rank 8 lattice. Let h be an element of this lattice with h h,hi = 2d. Then L2d = h ⊥ ∼ U 2 ⊥E 2 8 ⊥Zu with h u,ui = −2d is a lattice of signature (2,19) and we put